High-precision segmentation of the human cerebral cortex based on T1-weighted MRI is still a challenging task. When opting to use an intensity based approach, careful data processing is mandatory to overcome inaccuracies. They are caused by noise, partial volume effects and systematic signal intensity variations imposed by limited homogeneity of the acquisition hardware. We propose an intensity segmentation which is free from any shape prior. It uses for the first time alternatively grey (GM) or white matter (WM) based homogenization. This new tissue dependency was introduced as the analysis of 60 high resolution MRI datasets revealed appreciable differences in the axial bias field corrections, depending if they are based on GM or WM. Homogenization starts with axial bias correction, a spatially irregular distortion correction follows and finally a noise reduction is applied. The construction of the axial bias correction is based on partitions of a depth histogram. The irregular bias is modelled by Moody Darken radial basis functions. Noise is eliminated by nonlinear edge preserving and homogenizing filters. A critical point is the estimation of the training set for the irregular bias correction in the GM approach. Because of intensity edges between CSF (cerebro spinal fluid surrounding the brain and within the ventricles), GM and WM this estimate shows an acceptable stability. By this supervised approach a high flexibility and precision for the segmentation of normal and pathologic brains is gained. The precision of this approach is shown using the Montreal brain phantom. Real data applications exemplify the advantage of the GM based approach, compared to the usual WM homogenization, allowing improved cortex segmentation